3.189 \(\int \frac{x^4 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=160 \[ \frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{(20 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{19 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^5}-\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

-(d^3*(d - e*x)^4)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) + (19*d^2*(d - e*x)^3)/(15*e^5*
(d^2 - e^2*x^2)^(3/2)) - (6*d*(d - e*x)^2)/(e^5*Sqrt[d^2 - e^2*x^2]) - ((20*d -
e*x)*Sqrt[d^2 - e^2*x^2])/(2*e^5) - (19*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(
2*e^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.714212, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{6 d (d-e x)^2}{e^5 \sqrt{d^2-e^2 x^2}}-\frac{(20 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{19 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^5}-\frac{d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

-(d^3*(d - e*x)^4)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) + (19*d^2*(d - e*x)^3)/(15*e^5*
(d^2 - e^2*x^2)^(3/2)) - (6*d*(d - e*x)^2)/(e^5*Sqrt[d^2 - e^2*x^2]) - ((20*d -
e*x)*Sqrt[d^2 - e^2*x^2])/(2*e^5) - (19*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(
2*e^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 48.6786, size = 155, normalized size = 0.97 \[ - \frac{d^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 e^{5} \left (d + e x\right )^{4}} - \frac{19 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{5}} - \frac{12 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{e^{5} \left (d + e x\right )} + \frac{19 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{15 e^{5} \left (d + e x\right )^{3}} - \frac{4 d \sqrt{d^{2} - e^{2} x^{2}}}{e^{5}} + \frac{x \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

-d**3*(d**2 - e**2*x**2)**(3/2)/(5*e**5*(d + e*x)**4) - 19*d**2*atan(e*x/sqrt(d*
*2 - e**2*x**2))/(2*e**5) - 12*d**2*sqrt(d**2 - e**2*x**2)/(e**5*(d + e*x)) + 19
*d**2*(d**2 - e**2*x**2)**(3/2)/(15*e**5*(d + e*x)**3) - 4*d*sqrt(d**2 - e**2*x*
*2)/e**5 + x*sqrt(d**2 - e**2*x**2)/(2*e**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.183477, size = 98, normalized size = 0.61 \[ -\frac{285 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (448 d^4+1059 d^3 e x+713 d^2 e^2 x^2+75 d e^3 x^3-15 e^4 x^4\right )}{(d+e x)^3}}{30 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

-((Sqrt[d^2 - e^2*x^2]*(448*d^4 + 1059*d^3*e*x + 713*d^2*e^2*x^2 + 75*d*e^3*x^3
- 15*e^4*x^4))/(d + e*x)^3 + 285*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^5)

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 273, normalized size = 1.7 \[{\frac{x}{2\,{e}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{d}^{2}}{2\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{d}^{3}}{5\,{e}^{9}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}}+{\frac{19\,{d}^{2}}{15\,{e}^{8}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}}-10\,{\frac{d}{{e}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-10\,{\frac{{d}^{2}}{{e}^{4}\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ) }-6\,{\frac{d}{{e}^{7}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x)

[Out]

1/2/e^4*x*(-e^2*x^2+d^2)^(1/2)+1/2/e^4*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^
2*x^2+d^2)^(1/2))-1/5*d^3/e^9/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+19/
15/e^8*d^2/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-10/e^5*d*(-(x+d/e)^2*e
^2+2*d*e*(x+d/e))^(1/2)-10/e^4*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*
e^2+2*d*e*(x+d/e))^(1/2))-6/e^7*d/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*x^4/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.301768, size = 730, normalized size = 4.56 \[ \frac{15 \, e^{9} x^{9} - 150 \, d e^{8} x^{8} - 906 \, d^{2} e^{7} x^{7} + 270 \, d^{3} e^{6} x^{6} + 7827 \, d^{4} e^{5} x^{5} + 9500 \, d^{5} e^{4} x^{4} - 4180 \, d^{6} e^{3} x^{3} - 11400 \, d^{7} e^{2} x^{2} - 4560 \, d^{8} e x + 570 \,{\left (d^{2} e^{7} x^{7} + 7 \, d^{3} e^{6} x^{6} + 3 \, d^{4} e^{5} x^{5} - 31 \, d^{5} e^{4} x^{4} - 40 \, d^{6} e^{3} x^{3} + 12 \, d^{7} e^{2} x^{2} + 40 \, d^{8} e x + 16 \, d^{9} -{\left (d^{2} e^{6} x^{6} - 2 \, d^{3} e^{5} x^{5} - 19 \, d^{4} e^{4} x^{4} - 20 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} + 40 \, d^{7} e x + 16 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (15 \, e^{8} x^{8} - 15 \, d e^{7} x^{7} - 745 \, d^{2} e^{6} x^{6} - 4027 \, d^{3} e^{5} x^{5} - 3800 \, d^{4} e^{4} x^{4} + 6460 \, d^{5} e^{3} x^{3} + 11400 \, d^{6} e^{2} x^{2} + 4560 \, d^{7} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 3 \, d^{2} e^{10} x^{5} - 31 \, d^{3} e^{9} x^{4} - 40 \, d^{4} e^{8} x^{3} + 12 \, d^{5} e^{7} x^{2} + 40 \, d^{6} e^{6} x + 16 \, d^{7} e^{5} -{\left (e^{11} x^{6} - 2 \, d e^{10} x^{5} - 19 \, d^{2} e^{9} x^{4} - 20 \, d^{3} e^{8} x^{3} + 20 \, d^{4} e^{7} x^{2} + 40 \, d^{5} e^{6} x + 16 \, d^{6} e^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*x^4/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/30*(15*e^9*x^9 - 150*d*e^8*x^8 - 906*d^2*e^7*x^7 + 270*d^3*e^6*x^6 + 7827*d^4*
e^5*x^5 + 9500*d^5*e^4*x^4 - 4180*d^6*e^3*x^3 - 11400*d^7*e^2*x^2 - 4560*d^8*e*x
+ 570*(d^2*e^7*x^7 + 7*d^3*e^6*x^6 + 3*d^4*e^5*x^5 - 31*d^5*e^4*x^4 - 40*d^6*e^
3*x^3 + 12*d^7*e^2*x^2 + 40*d^8*e*x + 16*d^9 - (d^2*e^6*x^6 - 2*d^3*e^5*x^5 - 19
*d^4*e^4*x^4 - 20*d^5*e^3*x^3 + 20*d^6*e^2*x^2 + 40*d^7*e*x + 16*d^8)*sqrt(-e^2*
x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (15*e^8*x^8 - 15*d*e^7*x
^7 - 745*d^2*e^6*x^6 - 4027*d^3*e^5*x^5 - 3800*d^4*e^4*x^4 + 6460*d^5*e^3*x^3 +
11400*d^6*e^2*x^2 + 4560*d^7*e*x)*sqrt(-e^2*x^2 + d^2))/(e^12*x^7 + 7*d*e^11*x^6
+ 3*d^2*e^10*x^5 - 31*d^3*e^9*x^4 - 40*d^4*e^8*x^3 + 12*d^5*e^7*x^2 + 40*d^6*e^
6*x + 16*d^7*e^5 - (e^11*x^6 - 2*d*e^10*x^5 - 19*d^2*e^9*x^4 - 20*d^3*e^8*x^3 +
20*d^4*e^7*x^2 + 40*d^5*e^6*x + 16*d^6*e^5)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(x**4*sqrt(-(-d + e*x)*(d + e*x))/(d + e*x)**4, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.32269, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*x^4/(e*x + d)^4,x, algorithm="giac")

[Out]

Done